Existence and Hausdorff convergence of sliding phenomena in traveling waves for a fast-slow Keller-Segel equation

Abstract

A one-dimensional Keller-Segel equation with a piecewise-defined reaction term depending on the density gradient gives rise to a nonsmooth fast-slow system. Using Filippov convexification and geometric singular perturbation theory, we prove the existence of sliding traveling wave solutions for sufficiently small ε > 0 and characterize the corresponding sliding region. In the singular limit 𝜀 = 0 , we construct a singular sliding heteroclinic orbit on the critical manifold and show that it persists for sufficiently small ε > 0 under appropriate conditions on the wave speed and model parameters. The family of perturbed orbits converges to the singular orbit in the Hausdorff distance as ε → 0. These results provide a rigorous construction of sliding traveling waves for a chemotaxis model with gradient-dependent switching.